What is switching rows useful for?

I've learned about elementary row operations, there is one of them that seems a little bit weird to me: The row switching. It seems that a system of equations:

$$\begin{eqnarray*} {a_1x+b_1y+c_1z}&=&{d_1} \\ {a_2x+b_2y+c_2z}&=&{d_2} \\ {a_3x+b_3y+c_3z}&=&{d_3} \end{eqnarray*}$$

Can be represented by an augmented matrix:

$$\begin{pmatrix} {a_1}&{b_1}&{c_1}&{d_1}\\ {a_2}&{b_2}&{c_2}&{d_2}\\ {a_3}&{b_3}&{c_3}&{d_3} \end{pmatrix}$$

And then switching rows would ammount to just shift the order of the equations and hence, wouldn't change the solutions. This is a little confusing, I've been thinking about it's utility and until now, it seems that this ERO is just a visual aid for computing the other ERO's on matrices. Is that it or switching rows is something deeper and I still didn't notice?

For numerical algorithm, switching rows is called pivoting and improves stability compared to just "creating a $1$" in the row that happens to be in the current position. For manual / exact computations, there is absolutely no difference.
For a field $K$ (take $K = \mathbb{R}$ if you're confused by this), the group $GL_n(K)$ is generated by elementary matrices.
In a way, you might say that the nontriviality of the row switching operation corresponds to the fact that the matrix associated to a linear transformation depends not only on a choice of basis for the vector space, but also on the order of the basis vectors. Permuting the basis vectors corresponds to a nontrivial linear transformation, and the group of such permutations is $S_n$, which of course is generated by transpositions (corresponding to row-switches).